Let p be a prime number. We show that there is a one-to-one correspondence between the set of strongly nilpotent braces and the set of nilpotent pre-Lie rings of cardinality $p^{ n}$, for sufficiently large p. Moreover, there is an injective mapping from the set of left nilpotent pre-Lie rings into the set of left nilpotent braces of cardinality $p^{ n}$ for n+1<p. For the passage from pre-Lie rings to braces we use exactly the same method as suggested in [41].
Categories: math.RA